Range convergence monotonicity for vector measures and range monotonicity of the mass

Dekeyser, Justin;Van Schaftingen, Jean
(2020) Ricerche di Matematica — Vol. 69, n° 1, p. 293-326 (2020)

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We prove that the range of sequence of vector measures converging widely satisfies a weak lower semicontinuity property, hat the convergence of the range implies the strict convergence (convergence of the total variation) and that the strict onvergence implies the range convergence for strictly convex norms. In dimension 2 and for Euclidean spaces of any dimensions, we prove that the total variation of a vector measure is monotone with respect to the range.
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Dekeyser, J., & Van Schaftingen, J. (2020). Range convergence monotonicity for vector measures and range monotonicity of the mass. Ricerche di Matematica, 69(1), 293-326. https://doi.org/10.1007/s11587-019-00463-x (Original work published 2020)